F-Droid

I recommend F-Droid Basic, it is more modern.

You just copy the repo URLs and install the other client and add the repos there.

Android apps are handled by the android package manager. That uses the signatures that were used to sign APKs. The stores just download them.

The source of the APKs is important though. F-Droid builds them on its own, this results in a more consistent build that may be more secure. But you need to trust F-Droid AND the developer.

Also, F-Droid is often multiple versions behind, where the developers might release newer versions.

F-droid uses a single key to sign all APKs, normally every dev has their own key.

If you use Obtainium, you normally get the APKs directly from releases of the devs. The same goes if you have external repositories, btw I updated this list recently and there are a lot of EOL repos.

Once an APK is installed, you can only update it, if the key matches. This means if you want to switch for example from Mull (F-Droid) to Mull (DivestOS repo) you need to reinstall the app.

But if you still use the apps from the same sources, you dont need to do that, and the client doesnt matter.